Linked Questions

1
vote
2answers
180 views

Addition of two algebraic integer numbers [duplicate]

Is the addition of two algebraic integer numbers also algebraic? I, guess it is, but i can't prove it. I wonder if multiplication of them is also algebraic.
3
votes
0answers
258 views

Minimal polynomial for sum of algebraic numbers. [duplicate]

If I have two algebraic numbers $\alpha,\beta$ such that $A(\alpha) = 0$ and $B(\beta)=0$ where $A,B \in \mathbb{Q}[x]$ are the minimal polynomials of $\alpha$ and $\beta$ respectively. Knowing only ...
2
votes
1answer
138 views

If $\alpha$ and $\beta$ are roots of monic polynomials in $\mathbb{Z}[x]$, is $\alpha + \beta$? [duplicate]

If $\alpha$ and $\beta$ are roots of monic polynomials (not necessarily the same polynomial) in $\mathbb{Z}[x]$, is $\alpha + \beta$? I know that $\alpha + \beta$ will be the root of some monic ...
0
votes
0answers
58 views

Sum & Product of Two Algebraic Numbers [duplicate]

If you have algebraic numbers $x$ and $y$, and you know the polynomials of least degree of which each is a solution (written as a vectors of coefficients), then how is the vector of coefficients of ...
1
vote
0answers
29 views

Algebraic numbers theory [duplicate]

How I can prove that if there were $x$ and $y$ are algebraic numbers then $x+y$ and $x \cdot y$ are also algebraic numbers given that $x,y \in \mathbb R$?
36
votes
3answers
8k views

Enlightening proof that the algebraic numbers form a field

The proof I'm familiar with that the algebraic numbers $\mathbb A$ form a field uses the fact that the resultant of two polynomials $p,q\in\mathbb Q[x]$ satisfies the following properties: It is $0$ ...
10
votes
3answers
1k views

How to prove that algebraic numbers form a field?

I'd like to know how to prove algebraic numbers form a field, i.e., if $a,b$ are algebraic numbers, prove that $ab$ and $a+b$ are also algebraic numbers by finding specific polynomials that ...
15
votes
2answers
769 views

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
9
votes
1answer
271 views

How can I work with (i.e. add, multiply) algebraic numbers in practice?

I know that the real algebraic numbers $\mathbb A \subset \mathbb R$ form a field. I've seen this as a more theoretical result, but it's also seems nice idea to implement algebraic numbers for the ...
3
votes
1answer
360 views

a way to represent algebraic numbers in a computer

Say you want to represent the rational numbers in a computer. This is quite easy, you can think of them as pairs of integers. It is also easy to develop algorithms for adding, subtracting, multiplying ...
2
votes
1answer
302 views

Transcendence of $\sqrt{\pi}$

So it is known that $\pi$ is transcendental. With a little thought I was able to prove that $k\pi$ and $\pi^{k}$ for all $k\in\mathbb{Z}$ was transcendental. After that I thought about $\pi^{b}$ for ...
4
votes
3answers
122 views

construct polynomial from other polynomials

If I have a polynomial, P, with root $a$ and a polynomial, Q, with root $b$, is there a way to construct polynomial R such that $a+b$ is a root of R? Here's a concrete example. a = $\sqrt2$. $P(x) = ...
2
votes
4answers
169 views

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb{Q}$ (by finding a nonzero polynomial $p(x)$ with coefficients in $\mathbb{Q}$ which has $\sqrt[3] 2+\sqrt 5$ as a root). I first tried ...
0
votes
0answers
509 views

Set of algebraic integer form a ring.

An algebraic integer is a complex number that is the root of monic polynomial with integer coefficients. Show that the set of algebraic integers is a subring of $C$. (Hint: Use symmetric function ...
3
votes
3answers
86 views

Is $5^{1/5} - 3\cdot i$ algebraic?

I am studying the book Complex Variables with Applications written by Herb Silverman. In this book, problem number 8 in Question 1.7 is as in the following. Is $5^{1/5} - 3\cdot i$ algebraic? (i.e, ...

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